Invertible matrix

An invertible matrix is a square matrix whose product with another matrix gives the identity matrix. Its inverse is unique.

Matrix inverse: The unique matrix that multiplies a given invertible matrix to produce the identity matrix. This is the matrix that makes the defining product equal the identity.
Square matrix: A matrix with the same number of rows and columns. Only square matrices have two-sided multiplicative inverses of the same size.
System of linear equations: A collection of linear equations sharing the same variables. An invertible coefficient matrix guarantees a unique solution for every right-hand side.
Singular matrix: A square matrix that has no multiplicative inverse. It is the direct opposite: its determinant is zero and its transformation is not bijective.
Rank–nullity theorem: A theorem equating a linear map’s domain dimension with the sum of its rank and nullity. For a square invertible matrix, full rank forces nullity zero.
Identity matrix: A square matrix with ones on the main diagonal and zeros elsewhere, acting as the multiplicative identity. An inverse is defined by multiplying to this matrix.
Matrix multiplication: An operation that combines compatible matrices by multiplying rows of one by columns of another. Invertibility is defined through matrix multiplication.
Change of basis: A conversion between coordinate descriptions of the same vector or linear transformation. The coordinate conversion matrix must be invertible so the conversion can be reversed.
Pseudoinverse: A generalized inverse defined for rectangular or singular matrices, commonly the Moore–Penrose pseudoinverse. It extends useful inverse-like operations where an ordinary inverse does not exist.
Linear isomorphism: A bijective linear map between vector spaces, whose inverse is also linear. An invertible matrix represents an isomorphism in coordinates.

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